Finite Group DatabasebetaFinite Group DatabasebetaOrder 1248.
292 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D13 |
| commutator, socle, cosocle | C2×C3×C13 | C24 |
| Fitting | C3×Q8×C13 | C22 |
| hypercenter | Q8 | D3×D13 |
| hypocenter | C3×C13 | 2−1+4 |
| LCS | G ▷ C2×C3×C13 ▷ C3×C13 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×D39 | C2 | semidirect |
| 2. D3×Q8×C13 | C2 | semidirect |
| 3. C3×#6131 | C2 | semidirect |
| 4. #10597737 | C2 | semidirect |
| 5. #10597732 | C2 | semidirect |
| 6. #10597731 | C2 | semidirect |
| 7. Dic6×D13 | C2 | semidirect |
| 23. #6131 | D3 | semidirect |
| 61. D3×Q8 | D13 | semidirect |
| 66. C3×C13 | 2−1+4 | semidirect |
| 91. C13 | #1010 | semidirect |
| 122. C3 | #88254 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D39 | 2 |
| D3×Q8×C13 | 2 |
| C3×#6131 | 2 |
| #10597737 | 2 |
| #10597732 | 2 |
| #10597731 | 2 |
| Dic6×D13 | 2 |
| #88254 | 3 |
| #1010 | 13 |